In what follows we recall some limit theorems for continuous local martingales. We use these limit theorems for studying the asymptotic behaviour of the MLE of b. First we recall a strong law of large numbers for continuous local martingales.
C.1 Theorem. (Liptser and Shiryaev [36, Lemma 17.4]) Let Ω,F,(Ft)t∈R+,P
be a filtered probability space satisfying the usual conditions. Let (Mt)t∈R+ be a square-integrable continuous local martingale with respect to the filtration (Ft)t∈R+ such that P(M0 = 0) = 1. Let (ξt)t∈R+ be a progressively measurable process such that
P Z t
0
ξu2dhMiu <∞
= 1, t∈R+, and
Z t
0
ξu2dhMiu a.s.
−→ ∞ as t→ ∞, (C.1)
where (hMit)t∈R+ denotes the quadratic variation process of M. Then Rt
0ξudMu Rt
0ξ2udhMiu
−→a.s. 0 as t→ ∞. (C.2)
If (Mt)t∈R+ is a standard Wiener process, the progressive measurability of (ξt)t∈R+ can be relaxed to measurability and adaptedness to the filtration (Ft)t∈R+.
The next theorem is about the asymptotic behaviour of continuous multivariate local martingales, see van Zanten [46, Theorem 4.1].
C.2 Theorem. (van Zanten [46, Theorem 4.1]) Let Ω,F,(Ft)t∈R+,P
be a filtered probability space satisfying the usual conditions. Let (Mt)t∈R+ be a d-dimensional square-integrable continuous local martingale with respect to the filtration (Ft)t∈R+ such that P(M0 = 0) = 1. Suppose that there exists a function Q:R+→Rd×d such that Q(t) is an invertible (non-random) matrix for all t∈R+, limt→∞kQ(t)k= 0 and
Q(t)hMitQ(t)⊤ −→P ηη⊤ as t→ ∞,
where η is a d×drandom matrix. Then, for eachRk-valued random vector v defined on (Ω,F,P), we have
(Q(t)Mt,v)−→D (ηZ,v) as t→ ∞,
where Z is a d-dimensional standard normally distributed random vector independent of (η,v).
We note that Theorem C.2 remains true if the function Q is defined only on an interval [t0,∞) with some t0 ∈R++.
Acknowledgements
We would like to thank the referee for his/her comments that helped us to improve the paper.
References
[1] Alfonsi, A., (2005). On the discretization schemes for the CIR (and Bessel squared) processes.
Monte Carlo Methods and Applications 11(4)355–384.
[2] Barczy, M., Ben Alaya, M., Kebaier, A. and Pap, G. (2016). Asymptotic behav-ior of maximum likelihood estimators for a jump-type Heston model. Available on ArXiv:
http://arxiv.org/abs/1509.08869
[3] Barczy, M., D¨oring, L., Li, Z. and Pap, G. (2013). On parameter estimation for critical affine processes. Electronic Journal of Statistics 7 647–696.
[4] Barczy, M., Li, Z. and Pap, G. (2015). Yamada–Watanabe results for stochastic differential equations with jumps. International Journal of Stochastic Analysis Volume 2015 Article ID 460472, 23 pages.
[5] Barczy, M., Li, Z. and Pap, G. (2015). Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA. Latin American Journal of Probability and Mathematical Statistics.12(1) 129–169.
[6] Ben Alaya, M. and Kebaier, A.(2012). Parameter estimation for the square root diffusions:
ergodic and nonergodic cases. Stochastic Models 28(4)609–634.
[7] Ben Alaya, M. and Kebaier, A. (2013). Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions.Stochastic Analysis and Applications 31(4) 552–573.
[8] Billingsley, P. (1999). Convergence of probability measures, 2nd ed. John Wiley & Sons, Inc., New York.
[9] Bhattacharya, R. N.(1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes.Zeitschrift f¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete 60 185–201.
[10] Cox, J. C.,Ingersoll, J. E.andRoss, S. A.(1985). A theory of the term structure of interest rates. Econometrica 53(2)385–407.
[11] Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes.
The Annals of Probability 34(3)1103–1142.
[12] Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, California.
[13] Duffie, D. and Gˆarleanu, N. (2001). Risk and valuation of collateralized debt obligations.
Financial Analysts Journal 57(1) 41–59.
[14] Duffie, D.,Filipovi´c, D.andSchachermayer, W.(2003). Affine processes and applications in finance.Annals of Applied Probability 13984–1053.
[15] Feller, W. (1951). Two singular diffusion problems.Annals of Mathematics 54(1)173–182.
[16] Filipovi´c, D. (2001). A general characterization of one factor affine term structure models.
Finance and Stochastics 5(3) 389–412.
[17] Filipovi´c, D., Mayerhofer, E.and Schneider, P.(2013). Density approximations for mul-tivariate affine jump-diffusion processes. Journal of Econometrics 176(2) 93–111.
[18] Fu, Z. and Li, Z.(2010). Stochastic equations of non-negative processes with jumps.Stochastic Processes and their Applications 120 306–330.
[19] Huang, J., Ma, C. and Zhu, C. (2011). Estimation for discretely observed continuous state branching processes with immigration. Statistics and Probability Letters 811104–1111.
[20] Ikeda, N.andWatanabe, S. (1989).Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland/Kodansha, Amsterdam/Tokyo.
[21] Jacod, J. and M´emin, J. (1976). Caract´eristiques locales et conditions de continuit´e absolue pour les semi-martingales. Zeitschrift f¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete,35 1–37.
[22] Jacod, J. and Protter, P.(2012). Discretization of processes, Springer, Heidelberg.
[23] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed.
Springer-Verlag, Berlin.
[24] Jiao, Y., Ma, C.andScotti, S.(2016). Alpha-CIR model with branching processes in sovereign interest rate modelling. Available on ArXiv: http://arxiv.org/abs/1602.05541
[25] Jin, P., R¨udiger, B. and Trabelsi, C. (2016). Exponential ergodicity of the jump-diffusion CIR process. In: Stochastics of environmental and financial economics—Centre of Advanced Study, Oslo, Norway, 2014–2015, pages 285–300, Springer Proceedings in Mathematics & Statis-tics, 138, Springer, Cham.
[26] Keller-Ressel, M.(2008). Affine Processes - Theory and Applications in Finance.PhD Thesis, Vienna University of Technology, pages 110.
[27] Keller-Ressel, M. (2011). Moment explosions and long-term behavior of affine stochastic volatility models. Mathematical Finance 21(1)73–98.
[28] Keller-Ressel, M. and Mijatovi´c, A.(2012). On the limit distributions of continuous-state branching processes with immigration. Stochastic Processes and their Applications 122 2329–
2345.
[29] Keller-Ressel, M.and Steiner, T.(2008). Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance and Stochastics 12(2)149–172.
[30] K¨uchler, U.and Sørensen, M.(1997).Exponential families of stochastic processes, Springer-Verlag, New York.
[31] Kyprianou, A. E. (2014).Fluctuations of L´evy Processes with Applications, 2nd ed. Springer-Verlag, Berlin Heidelberg.
[32] Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Fi-nance. Chapman & Hall/CRC.
[33] Li, Z. (2011).Measure-Valued Branching Markov Processes. Springer-Verlag, Heidelberg.
[34] Li, Z. (2012). Continuous-state branching processes. Available on ArXiv:
http://arxiv.org/abs/1202.3223
[35] Li, Z. and Ma, C. (2013). Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model. Stochastic Processes and their Applications 125(8) 3196–3233.
[36] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes II. Applications, 2nd edition. Springer-Verlag, Berlin, Heidelberg.
[37] Luschgy, H. (1992). Local asymptotic mixed normality for semimartingale experiments. Prob-ability Theory and Related Fields 92151–176.
[38] Luschgy, H. (1994). Asymptotic inference for semimartingale models with singular parameter points. Journal of Statistical Planning and Inference 39 155–186.
[39] Ma, R. (2013). Stochastic equations for two-type continuous-state branching processes with immigration. Acta Mathematica Sinica, English Series 29(2)287–294.
[40] Mai, H.(2012). Drift estimation for jump diffusions: time-continuous and high-frequency obser-vations. Ph.D. Dissertation. Humboldt-Universit¨at zu Berlin.
[41] Overbeck, L. (1998). Estimation for continuous branching processes. Scandinavian Journal of Statistics 25(1)111–126.
[42] Overbeck, L.andRyd´en, T.(1997). Estimation in the Cox-Ingersoll-Ross model.Econometric Theory 13(3) 430–461.
[43] Pinsky, M. A. (1972). Limit theorems for continuous state branching processes with immigra-tion. Bulletin of the American Mathematical Society 78(2)242–244.
[44] Sato, K.-I. (1999).L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.
[45] Sørensen, M.(1991). Likelihood methods for diffusions with jumps. In: N. U. Prabhu and I.V.
Basawa, Eds., Statistical Inference in Stochastic Processes, Marcel Dekker, New York, 67–105.
[46] van Zanten, H. (2000). A multivariate central limit theorem for continuous local martingales.
Statistics & Probability Letters 50(3)229–235.